Question: $\dfrac{d}{dx}\left(\sqrt{x^5}\right)=$
Solution: The strategy We can first rewrite the radical as a rational power of $x$. Then, the derivative can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ (Remember that this applies even when $n$ is a fraction.) Rewriting the radical as a rational power $\sqrt{x^5}=x^{^{\frac{5}{2}}}$ Differentiating using the power rule $\begin{aligned} &\phantom{=}\dfrac{d}{dx}\left(x^{^{\frac{5}{2}}}\right) \\\\ &=\dfrac{5}{2}x^{^{\frac{5}{2}-1}} \gray{\text{The power rule}} \\\\ &=\dfrac52x^{^{\frac{3}{2}}} \end{aligned}$ In conclusion, we found that $\dfrac{d}{dx}\left(\sqrt{x^5}\right)=\dfrac52x^{^{\frac{3}{2}}}$. This can also be written as $2.5\sqrt{x^3}$ (all equivalent forms are accepted).